Theorem 7.8 
[1]} Let \(U\)  be Haar distributed on \(\mathrm {U}(N)\) . Let \({\bf a}=(a_1,\dots , a_k)\) , \({\bf b}=(b_1,\dots , b_k)\) , with \(a_i, b_i \in \mathbb {Z}_{\ge 0}\) . Then for \(N\ge \max \left\lbrace \sum _{j=1}^ka_jj, \sum _{j=1}^kb_jj\right\rbrace \) ,
\(\mathbb {E}\left(\prod _{j=1}^k (\mathrm {Tr}(U^j))^{a_j} \overline{(\mathrm {Tr} (U^j))}^{b_j} \right)=\delta _{\bf a b} \prod _{j=1}^n j^{a_j} a_j!\) 
